Problem 26 - Precalculus with Limits

Question

In Exercises 25-32, find all (a) minors and (b) cofactors of the matrix. \(\left[ \begin{array}{r} 0 & 10 \\ 3 & -4 \end{array} \right]\)

Step-by-Step Solution

Answer

Minors are \(M_{1,1} = -4\), \(M_{1,2} = 3\), \(M_{2,1} = 10\), \(M_{2,2} = 0\). Cofactors are \(C_{1,1}= -4\), \(C_{1,2}= -3\), \(C_{2,1} = -10\), \(C_{2,2} = 0\).

Step 1: Calculate the Minors

The minors of a 2x2 matrix are relatively straightforward. This is because the minor of each element in the matrix is simply the other element that's not in the same row or column. Hence, \(M_{1,1} = -4\), \(M_{1,2} = 3\), \(M_{2,1} = 10\), \(M_{2,2} = 0\).

Step 2: Calculate the Cofactors

The cofactor \(C_{i,j}\) is given by \((-1)^{i+j}* M_{i,j}\). Thus, \(C_{1,1}= -4* (-1)^{1+1} = -4\), \(C_{1,2}= 3*(-1)^{1+2} = -3\), \(C_{2,1} = 10*(-1)^{2+1} = -10\), \(C_{2,2} = 0*(-1)^{2+2} = 0\).

Key Concepts

Understanding the 2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix, consisting of two rows and two columns. It is typically represented as \[\left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\], where each element in the matrix is an entry denoted by letters such as \(a\), \(b\), \(c\), and \(d\). This specific matrix configuration is straightforward and forms the basis for understanding more complex matrix operations. In a 2x2 matrix:

• The first row consists of elements \(a\) and \(b\).
• The second row is composed of elements \(c\) and \(d\).
• The columns are similarly defined, with the first column containing elements \(a\) and \(c\), and the second column containing \(b\) and \(d\).

These matrices are particularly useful in simple linear transformations and systems of equations. Comprehending how to manipulate the elements within a 2x2 matrix is fundamental in learning more advanced matrix operations.

Matrix Calculations: Minors and Cofactors

Matrix calculations such as finding minors and cofactors are vital tools in matrix algebra. These concepts are closely linked, aiding with computations such as determinants and inverses. A minor is the determinant of a smaller matrix obtained by removing one row and one column from the original matrix. For a 2x2 matrix, this is plainly the remaining element when one row and one column are excluded. In the given matrix, the four minors are

• \(M_{1,1} = -4\)
• \(M_{1,2} = 3\)
• \(M_{2,1} = 10\)
• \(M_{2,2} = 0\)

Cofactors are derived from minors, with the role of adjusting the sign based on its position. The position is determined by the formula \((-1)^{i+j} \), where \(i\) and \(j\) represent the row and column of the element. This ensures the right sign application for each cofactor. For the provided matrix, the cofactors are:

• \(C_{1,1} = -4\)
• \(C_{1,2} = -3\)
• \(C_{2,1} = -10\)
• \(C_{2,2} = 0\)

Understanding these concepts makes solving larger matrices much easier.

Determinants in 2x2 Matrices

Determinants are scalar values that provide insight into properties of a matrix, such as whether it is invertible or not. The determinant is especially easy to compute in the 2x2 matrix case, based on the elements within. For any 2x2 matrix \[\left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\], the determinant \(det(A)\) is calculated as:\[det(A) = ad - bc\]When this value is zero, the matrix does not have an inverse and is said to be singular. For our example matrix \[\left[ \begin{array}{cc} 0 & 10 \ 3 & -4 \end{array} \right]\], its determinant can be found by applying the formula:\[det(A) = (0)(-4) - (10)(3) = 0 - 30 = -30\]Since the determinant is not zero, the matrix is invertible. Working with determinants gives valuable information about systems represented by matrices, and they serve as a crucial concept in both theoretical and applied mathematics.